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Computing with physics
From biologica l to a rtificia l intelligenceand back aga in
Mihai A. Petrovici
Electronic Vision(s)Kirchhoff Institute for Physics
University of Heidelberg
Computa tiona l Neuroscience GroupDepartment of Physiology
University of Bern
Sampling-based Bayesian computa tion
Petrovici & Bill et a l. (2013, 2016), Leng & Petrovici et a l. (2016, 2018), Kungl et a l. (in prep.)
Whence stochasticity?
Stochasticity in (dedica ted) discrete systems
Stochasticity in continuous systemsInjection of stochasticity
Embedded stochasticity
Dold & Bytschok & Petrovici et a l. (2018)
Stochasticity from function: Bayesian inference & dreaming
Dold & Bytschok & Petrovici et a l. (2018), Kungl et a l. (in prep.)
Physica l stochastic computa tion without noise
Leng & Petrovici et a l. (2016, 2018), Zenk (2018), Kungl et a l. (in prep.)
Superior mixing in spiking networks
Biologica l mechanisms for superior mixing
Leng & Petrovici et a l. (2016, 2018), Korcsak-Gorzo & Baumbach & Petrovici et a l., in prep.
E z
cortica l oscilla tions
short-term synaptic plasticity
What’s wrong with Euclidean gradient descent?
Δ𝑤𝑤syn = −𝜂𝜂𝜕𝜕𝜕𝜕 𝛼𝛼𝑤𝑤syn
𝜕𝜕𝑤𝑤syn = −𝜂𝜂𝛼𝛼𝜕𝜕𝜕𝜕 𝑤𝑤syn
𝜕𝜕𝑤𝑤syn
𝛻𝛻n = 𝐺𝐺(𝒘𝒘)−1𝛻𝛻e
Kreutzer et a l., in prep.
Synaptic plasticity as na tura l gradient descent
loca l lea rning:Δ𝑛𝑛𝑤𝑤 = 𝜂𝜂 ∫0𝑇𝑇 𝛾𝛾0 𝑦𝑦∗ − 𝜙𝜙 𝑉𝑉 𝜙𝜙′ 𝑉𝑉
𝜙𝜙 𝑉𝑉𝒙𝒙𝜖𝜖
𝒓𝒓− 𝛾𝛾1 + 𝛾𝛾2𝒘𝒘 𝑑𝑑𝑑𝑑
Chen et a l. (2013)Häusser et a l. (2001) Loewenstein et a l. (2011)
Kreutzer et a l., in prep.
Hierarchica l networks and the credit-assignment problem
propagate and assign error
reduce cost (gradient descent)
?
Lagrangian mechanics
fundamenta l principle in→ mechanics→ geometrica l optics→ electrodynamics→ quantum mechanics→ …→ neurobiology?
principle of (least) sta tionary action
𝛿𝛿 �𝑑𝑑𝑑𝑑 𝐿𝐿 𝒒𝒒, �̇�𝒒 = 0
𝜕𝜕𝐿𝐿𝜕𝜕𝑞𝑞𝑖𝑖
−𝑑𝑑𝑑𝑑𝑑𝑑
𝜕𝜕𝐿𝐿𝜕𝜕�̇�𝑞𝑖𝑖
= 0
Euler-Lagrange equations of motion
⇒
𝐸𝐸 𝒖𝒖 = �𝑖𝑖
12𝒖𝒖𝑖𝑖 −𝑾𝑾𝑖𝑖�𝒓𝒓𝑖𝑖−1 2 + 𝛽𝛽
12𝒖𝒖𝑁𝑁 − 𝒖𝒖𝑁𝑁
tgt 2
𝜕𝜕𝐿𝐿𝜕𝜕�𝒖𝒖𝑖𝑖
−𝑑𝑑𝑑𝑑𝑑𝑑
𝜕𝜕𝐿𝐿𝜕𝜕�̇𝒖𝒖𝑖𝑖
= 0
�̇�𝑾𝑖𝑖 = −𝜂𝜂𝜕𝜕𝐸𝐸𝜕𝜕𝑾𝑾𝑖𝑖
𝜏𝜏�̇�𝒖𝑖𝑖 = 𝑾𝑾𝑖𝑖𝒓𝒓𝑖𝑖−1 − 𝒖𝒖𝑖𝑖 + 𝒆𝒆𝑖𝑖 → neuron dynamics!
�𝒆𝒆𝑖𝑖 = �𝒓𝒓𝑖𝑖′ 𝑾𝑾𝑖𝑖+1𝑇𝑇 𝒖𝒖𝑖𝑖+1 −𝑾𝑾𝑖𝑖+1�𝒓𝒓𝑖𝑖 → error backprop!
�̇�𝑾𝑖𝑖 = 𝜂𝜂 𝒖𝒖𝑖𝑖 −𝑾𝑾𝑖𝑖�𝒓𝒓𝑖𝑖−1 �𝒓𝒓𝑖𝑖−1𝑇𝑇 → Urbanczik-Senn learning rule!
⇒
1
𝑖𝑖
𝑖𝑖 − 1
𝑁𝑁
⋮
⋮
Dold et a l. (Cosyne abstracts 2019, paper in prep.)
Lagrangian mechanics for neuronal networks
𝒖𝒖 = �𝒖𝒖 − 𝜏𝜏�̇𝒖𝒖𝐿𝐿 �𝒖𝒖, �̇𝒖𝒖
energy 𝐸𝐸
𝜷𝜷 = 𝟎𝟎, �̇�𝑾 = 𝟎𝟎→ 𝑬𝑬 = 𝟎𝟎
𝜷𝜷 ≠ 𝟎𝟎, �̇�𝑾 = 𝟎𝟎→ 𝑬𝑬 ≠ 𝟎𝟎
𝜷𝜷 ≠ 𝟎𝟎, �̇�𝑾 ≠ 𝟎𝟎→ 𝑬𝑬 = 𝟎𝟎
Sacramento et a l. (2017), Dold et a l. (Cosyne abstracts 2019, paper in prep.)
Biophysica l implementa tion
𝒓𝒓 𝒕𝒕 ≈ 𝝋𝝋(𝒖𝒖 𝒕𝒕 + 𝝉𝝉 )
• loca l representa tion of errors for plausible synaptic plasticity
• prospective coding for continuous dynamics